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Eggs in finite projective spaces and unitals in translation planes

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Abstract

Inspired by the connection between ovoids and unitals arising from the Buekenhout construction in the André/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of \(\textrm{PG}(4m-1,q)\), \(m\ge 1\), are a generalization of ovoids, we explore the relation between eggs and unitals in translation planes of higher dimension over their kernel. By investigating such a relationship, we construct a unital in the Dickson semifield plane of order \(3^{10}\), which is represented in \(\textrm{PG}(20,3)\) by a cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in \(\textrm{PG}(19,3)\). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane.

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Authors and Affiliations

  1. Dipartimento di Matematica, Informatica ed Economia, Università degli Studi della Basilicata, Viale dell’Ateneo Lucano 10, 85100, Potenza, Italy

    Alessandro Siciliano

  2. School of Mathematical Sciences, The University of Adelaide, Adelaide, SA, 5005, Australia

    Tim Penttila

  3. UP FAMNIT, University of Primorska, Koper, Slovenia

    Giusy Monzillo

Authors
  1. Giusy Monzillo
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  2. Tim Penttila
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  3. Alessandro Siciliano
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Correspondence to Alessandro Siciliano.

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Communicated by J. W. P. Hirschfeld

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The research was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM).

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Monzillo, G., Penttila, T. & Siciliano, A. Eggs in finite projective spaces and unitals in translation planes. Des. Codes Cryptogr. 91, 1475–1485 (2023). https://doi.org/10.1007/s10623-022-01162-9

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